Chapter 11.2, Problem 9E

Solution

To estimate: The area of the region above the $x$ -axis and under the graph of the given function from $x=0$ to $x=5$ .

The area of the region above the $x$ -axis and under the graph of the given function from $x=0$ to $x=5$ is $12.5$ square units.

Given information:

A curve and the value of $x$ ranges from $0$ to $5$ .

Formula used:

Area of rectangle $=l\times w$

Calculation:

The given curve is partitioned into strips of rectangles.

  

Area of each strip of rectangle can be calculated as below.

The area of first rectangle from left is given below. Width of the rectangle is $1$ and length of the rectangle is approximately $4$ .

Substitute $4$ for $l$ and $1$ for $w$ in the area of rectangle formula.

  $\begin{array}{c}\text{Area}=4\times 1\\ =4\end{array}$

Hence, the area of the first rectangle is $2$ square units.

The area of second rectangle from left is given below. Width of the rectangle is $1$ and length of the rectangle is approximately $5$ .

Substitute $5$ for $l$ and $1$ for $w$ in the area of rectangle formula.

  $\begin{array}{c}\text{Area}=5\times 1\\ =5\end{array}$

Hence, the area of the second rectangle is $5$ square units.

The area of third rectangle from left is given below. Width of the rectangle is $1$ and length of the rectangle is approximately $2$ .

Substitute $2$ for $l$ and $1$ for $w$ in the area of rectangle formula.

  $\begin{array}{c}\text{Area}=2\times 1\\ =2\end{array}$

Hence, the area of the third rectangle is $2$ square units.

The area of fourth rectangle from left is shown below. Width of the rectangle is $1$ and length of the rectangle is approximately $0.5$ .

Substitute $0.5$ for $l$ and $1$ for $w$ in the area of rectangle formula.

  $\begin{array}{c}\text{Area}=0.5\times 1\\ =0.5\end{array}$

Hence, the area of the fourth rectangle is $0.5$ square units.

The area of fifth rectangle from left is given below.

Substitute $2$ for $l$ and $1$ for $w$ in the area of rectangle formula.

  $\begin{array}{c}Area=2\times 1\\ =2\end{array}$

Hence, the area of the fifth rectangle is $2$ square units

Calculate the sum of areas of all the rectangles.

  $4+5+2+0.5+2=13.5$

Hence, the area of the region above the $x$ -axis and under the graph of the function from $x=0$ to $x=5$ is $13.5$ square units.